The Evolutionary Emergence of Intrinsic Regeneration in Artificial Developing Organisms Diego Federici Complex Adaptive Organically-inspired Systems group (CAOS) Norwegian University of Science and Technology N-7491 Trondheim, Norway Email: [email protected]

keywords: genetic algorithms, development, fault tolerance Abstract. Inspired upon the development of living systems, many models of artificial embryogeny are being proposed. These are usually aimed at the solution of some know limitations of evolutionary computation; among these scalability, flexibility and, more recently, fault-tolerance. This paper focuses on the latter, proposing an explanation of the intrinsic regenerative capabilities displayed by some models of multi-cellular development. Supported by the evidence collected from simulations, regeneration is shown to emerge as evolution converges to more regular regions of the genotype space. The conclusion is that intrinsic fault-tolerance emerges as evolution increases the evolvability of the development process.

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Introduction

Robustness to ‘hardware’ failures is a fundamental feature for living systems. Having to endure various sources of damage, such as injuries, aging, predators and parasites, organisms that display an endogenous resistance to external tampering and degradation have clearly an advantage when facing natural selection. It is therefore not surprising that, in order to support life, biological organisms naturally display a strong fault-tolerance. A possible path towards a good fault resistance is to design devices with some sort of functional redundancy, so that the negative effects of the loss of some components is mitigated by those which are still active. Multi-cellular living systems display an additional source of robustness derived by their regenerative capabilities. An example is provided by Hydras (Hydra Oligactis). Hidras can regenerate any damaged or dead cell, and severed body parts can even reconstruct the complete organism [1]. Cell regeneration is common also among more complex living organisms. For example the tail of the lizard and the limbs of the salamander regrow after being severed. In these cases, regeneration involves the production of highly differentiated tissues.

Notably, it has been shown that transplanted cells can assume specific roles based on the place where they are injected. For example, this technique allows mice to recover from spine injuries with the injection of staminal cells [2]. In these cases, fault-tolerance is based on the same ontogenetic processes which are the conrerstone of the organism’s development. For engineering purposes, devices that could automatically recover from faults are very appealing. Using a re-configurable substrate, we can envision systems that can heal themselves, effectively increasing their life-time without requiring any external support. Previous work conducted on artificial models of multi-cellular development has highlighted how this class of systems tends to display intrinsic regenerating properties. The term intrinsic refers to the fact that, albeit robustness can be boosted including it in the fitness function [3–5], recovery of phenotypic faults is also emergent, appearing as well when not selected for during evolution [3, 6, 7]. The fact that these systems are intrinsically fault-tolerant is very important: Even if is it possible to select individuals both on performance and robustness, testing all possible sources of faults can be computationally expensive (if not impossible). Since evolution tends to be very exploitative, faults that are not explicitly tested will most probably not be tolerated by the evolved designs. On the other hand, if a system presents some degree of intrinsic faulttolerance, we may expect that a necessarily partial robustness test will better generalize to unforeseen situations. Still, the fundamental reason that makes this class of embryogeny systems intrinsically fault-tolerant remains unclear. One may argue that, since the growth program and the variables it acts upon (i.e. cell types, chemicals, etc.) are distributed, development must necessarily provide a low sensitivity to phenotypic perturbations. Still, being based on rewrite rules, phenotype perturbations would be expected to induce a marked morphological divergence as faults get built upon. An explanation is offered by the canalization concept [8–10], i.e. that a canalized phenotype evolves to resist perturbations to its developmental process or its genotype: Robustness emerges because of the effects of a stabilizing selection. Simulations have shown that canalization emerges when developmental noise is present [11, 12]. These results are homologous to those presented in [3–5] where artificial developing organisms were selected for their phenotypic robustness . Canalization has also been shown to emerge with the evolution of genotypes with point-stable regulatory networks (independent of their function, [13]). This result is interesting because, as in [3, 6, 7], robustness is achieved in the absence of developmental noise, therefore without an explicit evolutionary advantage. In this paper we show that canalization and robustness can also emerge simply as a population of developing individuals evolves towards specific targets without developmental noise or the need of point-stable regulatory networks.

Evidence collected from simulations suggests how robustness is connected to a general evolutionary tendency to converge on stable genotype spaces (i.e. presenting a high degree of neutrality1 ). Altering the mutation rate, we prove how a more aggressive search produces individuals both with more robust phenotypes and converging towards wider neutral spaces. These results point out a subtle relationship between phylogeny and ontogeny, which does not appear to be explicitly dependant upon the user-defined selection criterion: The intrinsic fault-tolerance emerging in multi-cellular systems appears as a side-effect of the evolutionary preference for more regular regions of the genotype space.

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Related Work

Typically proposed to increase the scalability and flexibility of evolutionary computation, several indirected encoding schemes have been proposed. These ‘Artificial Embryogeny’ (AE, [14]) methods recursively construct the mature phenotype following the growth program defined in the genotype. Since selection operates at the level of the phenotype, the relationship between the evolving genotype and its inclusive fitness is mediated by the development process. This indirect path may trigger complex gene-to-gene interactions, which are captured by the concept of the Gene Regulatory Network (GRN). Since phenotypic maturation in AE is de facto a rewriting process, early models were based on grammar-based approaches in which the genotype defines the substitution rules which are repeatedly applied to the phenotype. Examples include the Matrix Rewriting scheme [15] and the Cellular Encoding [16]. Some models introduced additional contextual information in each rule definition [17, 18], so that phenotypic trait variations could be generated. Also, it is possible to implicitly define the grammar by means of an artificial GRN [19] and use the accumulated concentrations of simulated chemicals to modulate the characteristics of morphological constituents. In this direction, and inspired by Cellular Automata, a second approach is to evolve the rules by which cells alter their metabolism and duplicate. Cells are usually capable of sensing the presence of neighboring cells [20], releasing chemicals which diffuse in simulated 2D or 3D environments [21, 22], and moving and growing selective connections to neighboring cells [23]. Closely related to the one presented in this paper, the model proposed in [24] is based upon a fixed cartesian 2D lattice, a checkerboard, in which each cell occupies a given square. Artificial organisms are generated starting from a single cell. Every cell can replicate in the four cardinal directions taking the organism to maturation in a fixed number of development steps. All cells share the same genotype encoding the cell growth program (its regulatory network). In [24] the growth program is structured as a sequence 1

whose changes to the genotype produce none or little change to the phenotype/fitness

of rules. Rules are activated by matching the local neighborhood of a given cell and trigger specific cell responses: duplication, death and cell-state change. Individuals were evolved to produce tesselleting patterns. In [22], the growth program is represented by a boolean network. Cells belong to 1 of 4 different types and can release chemicals which undergo a simulated diffusion process. Specific evolutionary targets (2D patterns) were evolved and emergent self-healing dynamics were reported for the first time [6]. In [3] the previous model is extended with internal chemicals, which do not diffuse in the environment but are private to each cell. The growth program is encoded by a recursive neural network, and the organism’s genotype can contain several chromosomes, each one specifying a complete growth program. Individuals are initialized with a single chromosome which controls the entire development process. During evolution, additional chromosomes can be introduced by duplication (i.e. gene duplication [25]), each one being associated to a specific stage of development. By allowing several independent Embryonal Stages, this method proved capable of increasing overall evolvability in the evolution of specific 2D patterns, also showing a higher scalability then direct encoding. Also in this case, emergent fault-tolerance was reported. In [24, 22, 3], fitness was concerned only by the topological properties of mature individuals. In [5] the AE model in [22] was used to produce a 2-bit multiplier capable of recovering transient phenotype faults. In [4] the AE model in [3] was used to evolve a regenerating spiking neuro-controller for simulated Kephera robots. These last results prove the great potential that the evolution of complex fault-tolerant ontogenies can provide to the engineering community.

3

Methods: the Development Model

The AE model used in this paper is introduced in [3]. For clarity the model is explained in detail in this section. Phenotypes develop starting from a single cell placed in the center of a fixed size 2D checkerboard. Multi-cellular organisms reach maturation in a precise number of developmental steps (Nds ). Cells replicate and can release simulated chemicals in intra-cellular space (cell metabolism). Cell behaviour is governed by a growth program based on local variables, and represented by a simple recursive neural network (Morpher). 3.1

Cell State

Each position in the checkerboard can contain a cell, which is characterized by a state. The following table summarizes the information contained in each checkerboard position:

if a cell is present: the cell state: {active, passive} the cell type: an integer in [0, Nt − 1] the cell metabolism: a vector ∈ [−1, 1]Nm if no cell is present: observable cell type 0 In the simulations presented in this paper there is one metabolic chemical (Nm = 1) and the number of cell type (Nt ) is either 3 or 4. 3.2

The Regulatory System: the Morpher

Cell behavior is governed by an artificial neural network (Morpher) defined by the genotype. The Morpher’s inputs define the state of the regulatory system and its outputs encode the cell morphogenic actions. The Morpher input vector encodes the state of a particular cell (type and metabolism) and of the types of the 4 neighboring cells in the North, West, South and East directions (NWSE). At each developmental step, under the control of the Morpher outputs, existing active cells can change their own type, alter their metabolism and produce new cells. An active cell can also die or become passive. Each step, up to four new cells can be produced in any of the NWSE directions. In case, the mother cell specifies the daughter cells internal variables (type and metabolism) and whether they are active or passive. If necessary, existing cells are pushed sideways to create space for the new cells. When a cell is pushed outside the boundaries of the grid, it is permanently lost. The discrete cell type is encoded in a vector ∈ [−1, 1]Nd , in which each vector element is quantized to V values in the [-1,1] range. Therefore the number of cell types Nt equals V Nd . The input and output vectors are exemplified below: input neuron cell cell metabo- neighbors total vector bias age type lism cell types size 1 1 Nd Nm 4 × Nd 2 + Nm + 5Nd output change new cell new me- produce cell metabototal vector state? type tabolism cells? types lisms size 1 Nd Nm 4 4Nd 4Nm 5(Nd + Nm + 1)

Where: the cell age is set to 1 at cell birth and decays exponentially; ‘change state’ can take 4 values {no change, go passive, die, change type and metabolism}; ‘produce cells’ can take three values for each NWES direction {produce an active cell, produce a passive cell, do nothing}. Passive cells cannot replicate or change their own state. In the simulations presented herein, Nd = 1 and Nm = 1. The Morpher has 8 inputs, 15 outputs and contains no hidden layers. The genotype contains a floating point gene for each of the 120 Morpher weights.

3.3

Embryonal Stages

The regulatory system controls gene expression over two orthogonal dimensions: time and space. Development with Embryonal Stages (DES) implements a direct mechanism of Neutral Complexification for the temporal dimension. As development spans over several consecutive steps, the idea is to start evolution with a single growth program (chromosome/Morpher) which controls all the development steps. As evolution proceeds, a new chromosome can be added by gene duplication. The developmental steps are therefore partitioned into two groups/stages. The first, controlling the initial steps of embryogenesis, is associated with the old chromosome. The latter, completing growth, is associated with the new, identical, duplicated chromosome. Likewise, new chromosomes can be added one by one, each one controlling a partition of the last development steps. Being exact copies, new chromosomes do not alter development, and are therefore neutral. But eventual mutations can independently affect each duplicated gene. By unlocking the gene expression of different development phases, each chromosome can assume more specialized roles, de facto increasing the genotypic resolution around the area represented by the current mature phenotype. In fact, each new chromosome must take care of the maturation of an already partially developed phenotype. This new starting phenotype, as opposed to the zygote, is the result of the evolution of the previous chromosomes and hypothetically provides a flying start for the additional stage. Overall, the effect is an increase in genotype-phenotype correlation leading to higher evolvability [3]. In the simulations presented herein, only the chromosome associated to the latest stage is subjected to the evolutionary operators, while all other chromosomes remain fixed2 . 3.4

Evolutionary Details

Every population is composed of 400 individuals. The best 50 individuals are copied to the next generation and reproduce (elitism). Evolution comprises 1000 generations. The genotype contains a floating point number for each Morpher’s weight. Mutation takes each weight of the Morpher and adds to it Gaussian noise with 0 mean and Vmut variance (see Section 4 for actual values). With a .05 probability an offspring undergoes an additional symmetric mutation. The Morpher’s subnet responsible for the production of new cells in a chosen direction overwrites one or more of the other directions subnets. This operator should favor the evolution of phenotypes with various degrees of symmetry, but, since cells are not activated in parallel but follow a top-down left-to-right activation order, perfect symmetrical phenotypes usually require additional changes to the genotype. 2

In [3] it was shown that this restriction does not seem to affect the overall evolutionary dynamics while it speeds up the simulations

10% of the offspring are produced by crossover. Crossover exchanges all the weights connected to inherited outputs units. Organisms grow in a 32x32 checkerboard starting from a single active cell in position (16,16), with type 1 and metabolism 0. Development encompasses 12 development steps. At the end of a evolutionary run, genotypes comprise 12 embryonal stages (one chromosome for each development step). New stages are introduced every 1000/12 = 83.¯3 generations. Fitness Function Each cell in the mature phenotype is interpreted as a pixel, its color provided by the cell type. Fitness is proportional to the resemblance of an individual to a target pattern and is computed as shown in equation 1. For fitness computation, dead cells are assigned the default type 0 (black color) Fitness(P, T ) =

³P x,y

´ Equals ( P, T, x, y ) / ||T || ½

Equals ( P, T, x, y ) =

0 if P (x, y) 6= T (x, y) 1 if P (x, y) = T (x, y)

(1)

where P is the phenotype, T the target pattern. In case of ties, younger individuals are selected. Notice that in [3, 4], mechanisms devised to contrast premature convergence were present. In this case, we are less interested in evolvability and investigate the relation between phylogeny and ontogeny in the emergence of phenotypic regeneration. For this reason and clarity these mechanisms are not activated in the following simulations. Target Patterns The evolutionary targets are plotted in Figure 1.

4

Results

We analyse the results obtained from 36 independent runs with each parameter setting. We evolve populations whose individuals are selected in base of their resemblance to the targets plotted in Figure 1. After 1000 generations, the intrinsic fault-tolerance of the best individual of each run is tested. While during evolution, development is fault-free, during fault-tolerance tests, each cell at each development step is killed with a given probability (mortality rate). In case of death, cells are simply removed from the checkerboard. For each tested individual we compute: fitness recovery: the individual fitness score when subjected to faults. robustness: the phenotype stability to faults, i.e. a count of the phenotypic differences between the faulty and non-faulty individual averaged over the total number of cells.

Fig. 1. Evolutionary targets. On the left the 3-color 32x32 pattern (V = 3), on the right the 4-color 32x32 pattern (V = 4)

The latter is more indicative of the individuals’ intrinsic regenerative properties since computing only the recovered fitness score hides phenotypic changes that are neutral towards fitness. For simple combinatorial reasons, these are in fact more probable in less fit individuals. The averages of both indicators are plotted in Figure 2 for a 0.1 mortality rate. Populations with various levels of mutation variance have been evolved: Vmut = ei with i = {1, 0, −1, −2, −3, −4, −5, −6, −7} It is interesting to notice that, while performance appears maximized for intermediate values of Vmut , higher robustness emerges under stronger mutation rates. This intrinsic property of ontogeny appears mediated by the amount of phylogenetic variation. Since robustness is a feature which is not selected for, this result highlights a relationship between the domains of ontogeny and phylogeny. 4.1

Is Intrinsic Robustness Evolved?

We would like to know whether robustness is an emergent property of development in general, or more specifically it arises during evolution. In [3] it was reported that random individuals appeared less robust than fit ones. In Figure 3 we plot robustness over generations for all the best individuals from the best population evolved with Vmut = e−2 and 4 colors. It is observed that, similar to the results obtained in selection experiments [26, 27], robustness emerges after only a few generations. This shows how the most evolvable individuals also present an intrinsic fault-tolerance. Still robustness does not appear to be strictly proportional to fitness, as it also shown to decrease during evolution. This reflects the fact that robustness is neutral towards selection and its appearance is a byproduct of the evolutionary dynamics.

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Fig. 2. Emergent fault-tolerance with a 10% mortality rate: fitness and robustness averages over 100 tests. Tested are the fittest individuals of each population and mutation rate. Above individuals with 3 colors, below with 4 colors. Thin boxes display fitness without faults. Individuals were not selected for fault-tolerance

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Fig. 3. Intrinsic robustness levels during evolution. Averages over 100 tests for the best individuals of the highest scoring population with Vmut = e−2 and 4 colors. Robustness emerges during evolution but does not appear to be proportional to fitness. Thin boxes display fitness without faults.

For example, figure 4 shows how those individuals which are selected for reproduction are not those displaying the highest robustness. Still, the fact that the cluster of lower-fitness high-robustness individuals (labeled B, in Figure 4) is very dense may explain the reason behind the frequent emergence of faulttolerant individuals. 4.2

Relation between Robustness and Neutral Space

The presented results suggest a proportional relation between high mutation rates and phenotypic homeostasis. In this section, we argue that the emergent robustness of ontogeny is connected to the evolutionary preference for genotypes of high mutational robustness. According to the quasi-species model [28], apart from individuals of high fitness, selection would also prefer genotypes which are robust towards mutation. This is because with full replacement, those individuals which have more probability to produce viable fit offspring have a higher probability to survive as a quasi-species. As a result, populations tend to converge to genotype regions of higher neutrality (i.e. regions of Mutational Robustness, where fewer genotype mutations produce an observable phenotype/fitness change). Being more stable,

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Fig. 4. Scatter plot: intrinsic robustness over fitness. Averages over 100 tests for all the individuals of the last generation of the highest scoring population with Vmut = e−2 and 4 colors. Individuals laying on the right of the separation line will be selected for reproduction. The fittest individuals are clustered in a group (A), while the most robust ones are centered in another (B).

these regions are in fact attractors of the dynamic evolutionary system, see for example the analysis in [29]. In our case, we use a deterministic method without full replacement (elitism) to select the reproducing/surviving individuals, therefore the quasi-species model should not hold. In fact, individuals of highest fitness will always reproduce and survive no matter how rough is the fitness landscape around their genotype. Still, the quasi-species analysis is carried at equilibrium (as in [29]). Before reaching equilibrium, we argue, the lack of stochastic fluctuations in the surviving population is replaced by the effects of the randomic exploration of the search space. For example, let there be two distant genotype regions, R1 and R2 , so that individuals cannot migrate from one region to the other. R1 contains few individuals of optimal fitness FO and many of low fitness FL (i.e. a promising region but with a rough fitness landscape); while R2 contains few individuals of low fitness FL and many of high but sub-optimal fitness FSO , with FSO < FO (i.e. a less promising region but with regular fitness landscape).

Individuals laying on R1 have a low probability to generate fit individuals, while those laying on R2 have a statistically higher yield. The more frequent FSO solutions could take R1 individuals to extinction if FO solutions are not discovered in time. Therefore, even with elitism, the conclusions of the quasi-species model would hold, and evolution would push towards regions with more regular fitness landscapes, i.e. regions of higher mutational robustness. With a more aggressive mutation operator (higher Vmut ) we also expect populations to converge to regions of more marked neutrality.

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Fig. 5. Individual’s neutrality to genotype changes: phenotype variation per unit of genotype change. Averages over 48000 random genotype changes of the 5 best individuals of the 5 top-scoring populations with each parameter settings. Individuals evolved under stronger mutation rates (Vmut ) usually converge to wider neutral spaces.

We can test the neutrality of the genotype regions occupied by the best individuals of the 5 top scoring populations evolved with each parameter settings. For each tested individual, we alter the genotype and compute the corresponding phenotypic variation. Figure 5 shows the average phenotypic change per unit of genotype change. In the following tables, we report the maximum amplitude of the genotype alteration (G-distance3 ) which causes an average phenotype variation below the given threshold (averages over 48000 random genotype alterations of various amplitudes for each tested individual).

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measured as the euclidean distance in the 120-dimensional genotype space

3 colors G-distance for phenotype distance less then: average Vmut < 10−4 < 10−3 < 10−2 < 10−1 robustness (0.05) e1 0.20 4.50 14.00 ≥ 30.00 0.91 ± 0.00 e0 0.00 0.04 2.50 ≥ 30.00 0.89 ± 0.00 e−1 0.00 0.05 1.50 ≥ 30.00 0.90 ± 0.01 −2 e 0.00 0.01 0.10 18.00 0.86 ± 0.01 e−3 0.00 0.00 0.04 3.00 0.78 ± 0.01 e−4 0.00 0.00 0.04 2.50 0.62 ± 0.03 e−5 0.00 0.00 0.00 0.04 0.48 ± 0.00 e−6 0.00 0.00 0.00 0.01 0.35 ± 0.00 e−7 0.00 0.00 0.00 0.01 0.35 ± 0.00 4 colors G-distance for phenotype distance less then: average Vmut < 10−4 < 10−3 < 10−2 < 10−1 robustness (0.05) e1 0.07 0.40 9.00 ≥ 30.00 0.83 ± 0.01 e0 0.09 1.00 4.50 ≥ 30.00 0.87 ± 0.01 e−1 0.03 0.10 1.00 20.00 0.83 ± 0.01 e−2 0.01 0.06 0.40 6.00 0.76 ± 0.02 −3 e 0.00 0.00 0.02 1.50 0.70 ± 0.01 e−4 0.00 0.00 0.01 1.50 0.70 ± 0.00 e−5 0.00 0.00 0.00 0.70 0.49 ± 0.01 e−6 0.00 0.00 0.01 1.00 0.38 ± 0.01 e−7 0.00 0.00 0.00 0.03 0.30 ± 0.01 As expected there is a strong correlation between the mutation rate (Vmut ) and the average neutrality of the genotype changes. Notably, these results suggest that emergent fault-tolerance is connected to mutational robustness, pointing to a relation between the ontogenetic and phylogenetic domains. To confirm this hypothesis excluding the effects of fitness scores, we test individuals of equal fitness from the population of Figure 4. We take the most robust individual of the population and compare its neutral space size with the one of the least robust individual with the same fitness score. The most faulttolerant individual is shown to lay on a larger neutral space: 4 colors G-distance for phenotype distance less then: average fitness < 10−4 < 10−3 < 10−2 < 10−1 robustness (0.05) 0.5088 0.04 0.10 0.40 ≥ 30.00 .9829 ± .0004 0.5088 0.01 0.07 0.30 ≥ 30.00 .7073 ± .0081

5

Conclusions

Related work [22, 6, 3, 7] has show that some Artificial Embriogeny (AE) systems display emergent regenerative properties. With the aim to eventually produce as robust designs as those seen in nature, this tendency can be exploited to produce functional devices with remarkable fault-tolerance [4, 5].

Currently though, there is not a clear understanding of the reasons behind the emergence of the intrinsic robustness displayed by these development systems. It is often assumed that the distributed nature of ontogeny must play a fundamental role in the organism ability to recover from phenotypic faults. Still, logic also suggests that faults should propagate catastrophically as they are built upon during ontogeny. In this paper we address this issue linking the robustness to phenotypic faults occurring during development to the evolutionary tendency to converge on genotype spaces with a high degree of neutrality. Simulations conducted on the presented multi-cellular AE model have demonstrated that: R1: when the mutation rate is increased, evolution converges to more regular genotype regions, i.e. with a high degree of robustness to mutations. R2: a high robustness to mutations is related to a high degree of tolerance to phenotypic faults during development. These results are in good agreement with the canalization theory [8, 9], where the emergence of stabilizing selection originates by the evolutionary preference for regular genotype space regions. Additionally, if R1 fits well with the predictions derived by the quasi-species model [28], we will now argue that R2 is the logic extension of R1 in the case of ontogeny. In fact, the reason behind the evolutionary emergence of mutational robustness is that individuals converging to more regular fitness landscapes have a higher probability to flood a population with a single strain of related genotypes (i.e. a quasi-species). This implies that members of such successful quasi-species display a good neutrality to mutations: that when mutated phenotype/fitness changes are negligible. Without development, this can be achieved by reducing the negative effects of epistasis, see for example [30]. This is because, when mutations alter the genotype, a high level epistasis means that gene to gene interactions will amplify the phenotypic consequences of the change. With development, the rewriting process allows another path towards the achievement of a good neutrality to mutations: the possibility that a mutation causes a change to the growing phenotype which is cancelled (corrected) later on in development. Therefore a stable growth program is one that can also neutralize phenotypic variations. When evaluating the intrinsic fault-tolerance of an artificial embryogeny, we are in fact testing a facet of the genotype’s mutational robustness, its ability to suppress phenotypic variations caused by mutations. The conclusion is that fault-tolerance emerges during evolution because, as organisms compete to reach higher levels of evolvability, they converge to more regular (robust) genotype regions.

Future Work The fact that regeneration of multi-cellular systems emerges as a side effect of the optimization of the development process, allows us to draw two hypotheses: the first, that fault-tolerant developing organisms should be relatively easy to evolve; the second, that it should be possible to use faulttolerance to measure the evolvability of a development system. The former is already being validated by recent empirical results [4, 5]. The latter is being investigated, with the hope of producing a theory that would help the design of more evolvable artificial embryogenies. Acknowledgements I wish to thank Per Kristian Lehre for the many valuable discussions, and the anonymous reviewers for the quality of their feedback.

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